EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6643 ISSN 1307-5543 – ejpam.com Published by New York Business Global Soft Directed n-SuperHyperGraphs with Some Real-World Applications Takaaki Fujita1,∗, Florentin Smarandache2 1 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan. 2 University of New Mexico, Gallup Campus, NM 87301, USA. Abstract. Graph theory offers a powerful language for modeling pairwise relationships via vertices and edges [1]. Hypergraphs extend this framework by allowing hyperedges to join arbitrary subsets of vertices [2], and SuperHyperGraphs generalize further through iterative powerset constructions that capture multi-level hierarchical connectivity [3]. To address uncertainty and imprecision, a variety of set-based frameworks—fuzzy sets [4], soft sets [5], intuitionistic fuzzy sets [6], rough sets [7], neutrosophic sets [8], and plithogenic sets [9]—have been developed. In directed contexts, these ideas yield directed graphs [10], directed hypergraphs [11], directed SuperHyperGraphs [12], and soft directed graphs [13]. While Soft SuperHyperGraphs and Directed SuperHyperGraphs each capture important aspects of uncertainty and directionality in hierarchical networks, their integration remains largely unex- plored. In this paper, we introduce the Directed Soft SuperHyperGraph, a unified framework that combines directionality, recursive hyperstructure, and soft-set parameterization. We present for- mal definitions and core operations, demonstrate the model through real-world examples, and offer a qualitative comparison with existing approaches. Our framework is especially well suited to rep- resenting complex, multi-layered directed systems—such as urban infrastructure, transportation networks, and information-flow architectures—where deep hierarchies and uncertain relationships must be managed simultaneously. 2020 Mathematics Subject Classifications: 05C65 Key Words and Phrases: Superhypergraph, Directed Graph, Soft Graph, Soft Directed Super- HyperGraph, Hypergraph, Directed HyperGraph 1. Introduction 1.1. From Graphs to SuperHyperGraphs Simple graphs depict pairwise relationships by representing items as nodes and their connections as edges [14]. To describe interactions among multiple entities at once, hy- pergraphs permit each hyperedge to join any nonempty subset of vertices, thus capturing ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6643 Email addresses: Takaaki.fujita060@gmail.com (T.Fujita), smarand@unm.edu (F. Smarandache) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 2 of 27 higher-order associations [15, 16]. Building upon this, a SuperHyperGraph is formed by iteratively taking the powerset of the vertex set to create successive layers of links, em- bedding nested hierarchies within the network [17, 18]. 1.2. Capturing Uncertainty in Networks Real-world networks frequently include incomplete, imprecise, or even contradictory information. To model such ambiguity, a range of set-based frameworks—fuzzy sets [4], in- tuitionistic fuzzy sets [6], hyperfuzzy sets [19–21], hesitant fuzzy sets [22, 23], picture fuzzy sets[24], neutrosophic sets [25, 26], soft sets [5, 27], hypersoft sets[28–31], and plithogenic sets [32–34]—have been adapted for graph modeling. These adaptations give rise to struc- tures such as fuzzy graphs [35], intuitionistic fuzzy graphs [36], neutrosophic graphs [25], plithogenic graphs [37], and soft graphs [38], each integrating graded membership, inde- terminacy, and conflict measures to better reflect real-world uncertainty. Moreover, fuzzy graphs, soft graphs, and neutrosophic graphs have been generalized to hypergraphs and SuperHyperGraphs, with notable recent developments [39, 40]. 1.3. Introducing Directionality Many real-world systems exhibit directional or asymmetric ties. In a directed graph (digraph), each edge carries an orientation to indicate the direction of flow—be it infor- mation, resources, or influence—between nodes [41, 42]. This concept extends to directed hypergraphs, where hyperarcs distinguish between tail and head subsets, and further to di- rected superhypergraphs, which layer these directed connections across multiple hierarchical strata [12, 43–46]. 1.4. Contributions of This Paper From the foregoing discussion, research on Soft SuperHyperGraphs and Directed Su- perHyperGraphs is critically important, as it enables intuitive modeling of hierarchical real-world concepts. However, the integrated fusion of Soft SuperHyperGraphs and Di- rected SuperHyperGraphs has received little attention to date. To bridge this gap, in this paper we introduce the Directed Soft SuperHyperGraph, a novel framework that unifies di- rected relationships, hierarchical superhyperstructures, and soft-set parameterization. We demonstrate its effectiveness in modeling complex, multi-layered directed networks—such as urban transit systems, logistics networks, and information pipelines—where deep struc- tural hierarchies and uncertainty must be managed concurrently. 1.5. Structure of This Paper The remainder of this paper is organized as follows. In Section 2, we review the existing definitions of HyperGraphs, SuperHyperGraphs, the n-th iterated powerset, Soft Directed Hypergraphs, and related concepts. Section 3 presents the definition of the Directed Soft SuperHyperGraph and examines its fundamental properties. Section 4 provides a brief T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 3 of 27 discussion on the effectiveness of the Directed Soft SuperHyperGraph introduced in this paper. Finally, Section 5 concludes the paper and outlines directions for future research. 2. Preliminaries Throughout this paper, we work with finite, simple graphs unless stated otherwise. This section reviews the key definitions and notation that will be used in later sections; for more in-depth discussions, please refer to the cited sources. 2.1. SuperHyperGraphs Classical graphs model binary relations between vertices, but many applications re- quire connecting larger subsets at once. A hypergraph achieves this by replacing edges with hyperedges, each of which may join any nonempty collection of vertices [2, 47]. To capture multi-level or recursive structures, one can iterate the subset construction: a Su- perHyperGraph applies the powerset operation successively, creating “supervertices” and “superedges” at higher abstraction layers [3, 48–50]. Definition 1 (Base Set). [51] Let S be a base set, the foundational domain from which all subsequent constructions are drawn: S = {x : x is an element of the universe of discourse}. Definition 2 (Powerset). [52] The powerset of a set S, denoted POWS(S), is the collec- tion of all subsets of S, including ∅ and S itself: POWS(S) = {A : A ⊆ S}. Definition 3 (Hypergraph). [2, 47] A hypergraph H = (V,E) consists of • a finite vertex set V , • a family E of nonempty subsets of V , each called a hyperedge. Definition 4 (n-th Iterated Powerset). [53–55] Define the iterated powerset of a set X by POWS1(X) = POWS(X), POWSk+1(X) = POWS ( POWSk(X) ) , k ≥ 1. The corresponding nonempty iterated powerset is POWS∗ 1(X) = POWS(X) \ {∅}, POWS∗ k+1(X) = POWS ( POWS∗ k(X) ) \ {∅}. Example 1 (Organizational Hierarchy via Iterated Powersets). Let the base set of em- ployees be X = {Keisuke, Bob, Sakura}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 4 of 27 • POWS1(X) = POWS(X) is the set of all possible teams: {∅, {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {A,B,C}}. • POWS2(X) = POWS(POWS1(X)) is the set of all committees of teams, for example{ {{A,B}, {B,C}}, {{A}, {C}}, . . . } . • POWS3(X) = POWS(POWS2(X)) is the set of all councils of committees, such as{ {{{A,B}, {B,C}}, {{A}, {C}}}, . . . } . In practice: (i) A team is any subset of employees (level-1 grouping). (ii) A committee is a collection of teams (level-2 grouping). (iii) A council is a set of committees (level-3 grouping). Thus iterated powersets naturally model successive layers of organizational structure. Definition 5 (n-SuperHyperGraph). [56–58] Let V0 be a finite base set and define POWSk(V0) by iterating the powerset k times. An n-SuperHyperGraph is a pair SuHG(n) = (V,E), where V ⊆ POWSn(V0), E ⊆ POWSn(V0). Members of V are called n-supervertices and members of E are n-superedges. Example 2 (Software Architecture as a 2-SuperHyperGraph). Consider a software system whose lowest-level components are S = {Auth, DataAccess, UI}. At the first level, these components form modules: M1 = {Auth}, M2 = {DataAccess}, M3 = {UI}, so that POWS1(S) ⊇ V1 = {M1,M2,M3}. At the second level, modules combine into subsystems: SubA = {M1,M2}, SubB = {M2,M3}, SubAll = {M1,M2,M3}, so that POWS2(S) ⊇ V = {SubA,SubB, SubAll}. We choose the set of 2-superedges to capture key integration points: E = { e1 = { SubA,SubAll }, e2 = {SubB, SubAll } } , where: T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 5 of 27 • e1 models the integration between the Auth/Data subsystem and the full system. • e2 models the integration between the Data/UI subsystem and the full system. Hence SuHG(2) = (V, E ) is a 2-SuperHyperGraph representing the two–tier module-subsystem structure of the soft- ware architecture. 2.2. Soft SuperHypergraph Soft graphs capture families of subgraphs indexed by a parameter set, where each parameter selects a subgraph on a fixed vertex and edge base [38, 59, 60]. Equivalently, a soft graph realizes the concept of a soft set—a parameterized collection of subsets that models uncertainty or preference without requiring numeric membership or probability assignments [5, 27]. A soft hypergraph extends this notion by associating each parameter with a subset of vertices and the induced hyperedges within a hypergraph [61]. Finally, a soft n-superhypergraph applies these parameterized substructures to each layer of an n- SuperHyperGraph, yielding a hierarchically organized, uncertainty-aware model (cf.[12]). Definition 6 (Soft Hypergraph). [61, 62] Let H = (V,E) be a hypergraph with E ⊆ POWS(V ), and let C be a nonempty set of parameters. A soft hypergraph over H with parameters C is a quadruple ( H, C, A, B ) , where A : C −→ POWS(V ), B : C −→ POWS(E), and for each c ∈ C, B(c) ⊆ { e ∈ E | e ⊆ A(c)}. The pair ( A(c), B(c) ) is called the soft subhypergraph of H at parameter c. Example 3 (University Curriculum as a Soft Hypergraph). Consider the hypergraph H = (V,E) where V = {Calculus, Linear Algebra, Data Structures, Operating Systems, Networks}, E = { emath = {Calculus, Linear Algebra}, eCS = {Data Structures, Operating Systems, Networks}, ecomp = {Linear Algebra, Data Structures} } . Let C = {Math, CS, Inter} be the set of program-type parameters, and define A(Math) = {Calculus, Linear Algebra}, B(Math) = {emath}, A(CS) = {Data Structures, Operating Systems, Networks}, B(CS) = {eCS}, A(Inter) = {Linear Algebra, Data Structures}, B(Inter) = {ecomp}. Then ( H, C, A, B ) is a soft hypergraph in which each parameter c ∈ C induces the sub- hypergraph ( A(c), B(c) ) corresponding to the courses and curriculum bundles for that pro- gram type. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 6 of 27 Definition 7 (Soft n-Superhypergraph). [12] Let SuHG(n) = (V,E) be an n-SuperHyperGraph, and let C be a nonempty parameter set. A soft n-superhypergraph is a quintuple( V, E, C, A, B ) , where A : C −→ POWS(V ), B : C −→ POWS(E), such that for every c ∈ C, A(c) ⊆ V, B(c) ⊆ { e ∈ E | e ⊆ A(c)}. Thus ( A(c), B(c) ) is a sub-superhypergraph of SuHG(n) at parameter c. Example 4 (Personalized Curriculum Planning as a Soft 2-Superhypergraph). Imagine a university offering a modular curriculum. The ground set of all courses is S = {Calculus, Programming, Statistics, Logic}. At the first level, courses are grouped into subject areas (first-level supervertices): V1 = { MMath = {Calculus,Statistics}, MCS = {Programming,Logic} } ⊆ POWS1(S). At the second level, these areas form broader tracks (second-level supervertices): V2 = { T1 = {MMath}, T2 = {MCS}, TAll = {MMath,MCS} } ⊆ POWS2(S). We define the set of second-level “superedges” to indicate available program pathways: E = { e1 = {MMath, TAll}, e2 = {MCS, TAll} } , so that e1 and e2 represent progression from a subject area into the comprehensive track. Next, let the parameter set C capture two different student profiles: C = {Math-Focused, CS-Focused}. Define soft selection maps A : C −→ POWS(V2), B : C −→ POWS(E), by A(Math-Focused) = {T1, TAll}, B(Math-Focused) = {e1}, A(CS-Focused) = {T2, TAll}, B(CS-Focused) = {e2}. • Under the Math-Focused profile, a student selects the mathematics area first (T1) and then the joint track (TAll), following edge e1. • Under the CS-Focused profile, the programming–logic area (T2) leads into the joint track, via edge e2. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 7 of 27 Thus the quintuple ( V2, E, C, A, B ) is a soft 2-superhypergraph modeling how different student goals select distinct sub-paths through the curriculum hierarchy. Example 5 (Curriculum Program Selection as a Soft 3-Superhypergraph). Consider the ground set of courses S = {Calculus, Programming, Statistics, Logic}. First-level “modules” are MMath = {Calculus, Statistics}, MCS = {Programming, Logic}, POWS1(S) ⊇ {MMath, MCS}. Second-level “tracks” are T1 = {MMath}, T2 = {MCS}, TAll = {MMath, MCS}, POWS2(S) ⊇ {T1, T2, TAll}. Third-level “programs” (supervertices) are P1 = {T1}, P2 = {T2}, PAll = {T1, T2}, V = {P1, P2, PAll} ⊆ POWS3(S). Define the set of program hyperedges E = { e1 = {P1, PAll}, e2 = {P2, PAll} } ⊆ POWS(V ). Let the parameter set be C = {Math-Track, CS-Track}, and define the soft-selection maps A : C → POWS(V ), B : C → POWS(E) by A(Math-Track) = {P1, PAll}, B(Math-Track) = {e1}, A(CS-Track) = {P2, PAll}, B(CS-Track) = {e2}. Then ( V, E, C, A, B ) is a soft 3-superhypergraph modeling how students select and transition through programs in a three-tier curriculum. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 8 of 27 2.3. Soft Directed Hypergraphs A soft directed graph generalizes directed graphs by using parameters to select vertex subsets and their corresponding directed edge subsets [63–65]. A soft directed hypergraph generalizes both directed hypergraphs and soft sets by selecting substructures via param- eters (cf.[66, 67]). Definition 8 (Soft Directed Hypergraph). (cf.[66, 67]) Let SoDHG∗ = (Γ,Ξ) be a simple directed hypergraph, where Γ is the vertex set and Ξ ⊆ POWS(Γ) × POWS(Γ) is the set of hyperarcs (each an ordered pair of nonempty tail and head). Let C be a nonempty parameter set. A soft directed hypergraph is the quadruple( SoDHG∗, C, A, B ) , where A : C −→ POWS(Γ), B : C −→ POWS(Ξ), and for every c ∈ C, A(c) ⊆ Γ, B(c) ⊆ { (T,H) ∈ Ξ : T ⊆ A(c), H ⊆ A(c) } . Each pair ( A(c), B(c) ) is called the soft induced subhypergraph of SoDHG∗ at parameter c. Example 6 (Sensor Selection in a Directed Hypergraph). Consider a directed hypergraph SoDHG∗ = ({s1, s2, s3},Ξ) where Ξ = { ({s1}, {s2, s3}), ({s2}, {s3}) } . Let parameters C = {Day,Night}. Define A(Day) = {s1, s2}, B(Day) = {({s1}, {s2})}, A(Night) = {s2, s3}, B(Night) = {({s2}, {s3})}. Then ( A(c), B(c) ) for each c ∈ C is a directed subhypergraph on the active sensors, illus- trating how soft parameters select relevant hyperarcs. 2.4. Directed SuperHyperGraph Directed SuperHyperGraphs are graph classes that extend SuperHyperGraphs, re- spectively, in a manner analogous to Directed Graphs(cf.[3, 68]) . Below, we present their formal definitions and illustrative examples. Definition 9 (Directed n-SuperHyperGraph). (cf.[3, 68]) Let S be a nonempty base set and let n ≥ 0 be an integer. Define iterated powersets by POWS0(S) = S, POWSk+1(S) = POWS ( POWSk(S) ) (k ≥ 0). T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 9 of 27 A directed n-SuperHyperGraph is a pair DSuHG(n) = (V, E), where V ⊆ POWSn(S), E ⊆ POWSn(S) × POWSn(S), and each directed n-superedge e ∈ E is an ordered pair e = ( Tail(e), Head(e) ) , Tail(e), Head(e) ⊆ POWSn(S), typically both nonempty. Such an e carries “flow” from the entire set Tail(e) of n- supervertices into Head(e). Example 7 (Corporate Reporting as a Directed 2-SuperHyperGraph). Consider a small company with three employees: S = {Keisuke, Bob, Sakura}. They form two teams: TA = {Keisuke,Bob}, TB = {Bob, Sakura}, so that POWS1(S) ⊇ {TA, TB}. At the next level, these teams report to departments: D1 = {TA}, D2 = {TB}, DAll = {TA, TB}, so that POWS2(S) ⊇ V = {D1, D2, DAll}. We define the directed 2-superhyperedges E = { e1 = ( {D1}, {DAll} ) , e2 = ( {D2}, {DAll} )} . Here: • e1 models Team TA reporting up into the combined department DAll. • e2 models Team TB reporting up into DAll. Hence DSuHG(2) = ( V, E ) is a directed 2-SuperHyperGraph that captures the two-tier reporting structure in this organization. Example 8 (Corporate Hierarchy as a Directed 3-SuperHyperGraph). Consider a multi- national company with four employees: S = {Keisuke, Bob, Sakura, Dave}. They form two project teams: T1 = {Keisuke,Bob}, T2 = {Sakura,Dave}, T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 10 of 27 so that POWS1(S) ⊇ {T1, T2}. At the next level, these teams belong to departments: D1 = {T1}, D2 = {T2}, so that POWS2(S) ⊇ {D1, D2}. At the third level, departments are grouped into divisions: Div1 = {D1}, Div2 = {D2}, DivAll = {D1, D2}, so that POWS3(S) ⊇ V = {Div1,Div2,DivAll}. We then define the directed 3-superhyperedges E = { e1 = ( {Div1}, {DivAll} ) , e2 = ( {Div2}, {DivAll} )} . Here: • e1 models Division 1 reporting into the overall corporate division. • e2 models Division 2 reporting into the overall corporate division. Thus, DSuHG(3) = (V, E) is a directed 3-SuperHyperGraph capturing the four-tier hierarchy—employees, teams, de- partments, and divisions—of the organization. 3. Main Results In this section, we present the main contributions of this paper, focusing on the struc- ture and properties of Soft Directed n-SuperHyperGraphs. 3.1. Soft Directed n-SuperHyperGraphs We now introduce Soft Directed n-SuperHyperGraphs, which unite the ideas of directed n-SuperHyperGraphs and soft n-SuperHyperGraphs under a single framework. Definition 10 (Soft Directed n-SuperHyperGraph). Let S be a finite base set and n ≥ 0. Define iterated powersets by POWS0(S) = S, POWSk+1(S) = POWS ( POWSk(S) ) . A directed n-SuperHyperGraph is a pair DSuHG(n) = (V,E) with V ⊆ POWSn(S), E ⊆ POWSn(S) × POWSn(S), where each e ∈ E is an ordered pair (Tail(e),Head(e)). Let C be a nonempty parameter set. A soft directed n-SuperHyperGraph is the quadru- ple ( DSuHG(n), C, A, B ) , T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 11 of 27 where A : C −→ POWS(V ), B : C −→ POWS(E), and for each c ∈ C, A(c) ⊆ V, B(c) ⊆ { e ∈ E : Tail(e) ⊆ A(c), Head(e) ⊆ A(c)}. Each induced pair ( A(c), B(c) ) is called the soft directed sub-n-superhypergraph at pa- rameter c. Example 9 (Emergency Response Network as a Soft Directed 2-SuperHyperGraph). Con- sider a country-wide emergency response system. The base set of ground units is S = {TeamA, TeamB, TeamC}. They form regional clusters (first-level supervertices): V1 = { R1 = {TeamA,TeamB}, R2 = {TeamB,TeamC} } ⊆ POWS1(S). At the second level, these regions coordinate under a national command: V2 = { N = {R1, R2} } ⊆ POWS2(S). We define directed 2-superedges to model information flow: E = { e1 = ({R1}, {N}), e2 = ({R2}, {N}) } . Next, let the parameter set capture the system’s operational mode: C = {Routine, Emergency}. Define soft selection maps A : C −→ POWS(V2), B : C −→ POWS(E), by A(Routine) = {R1}, B(Routine) = {e1}, A(Emergency) = {R1, R2, N}, B(Emergency) = {e1, e2}. Here: • Under Routine mode, only region R1 reports upward, so we restrict to {R1} and edge e1. • Under Emergency mode, all regions and the national command are active, so we include R1, R2, N and both edges e1, e2. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 12 of 27 Thus the quadruple ( DSuHG(2), C, A, B ) is a soft directed 2-SuperHyperGraph, illustrating how parameter settings select which supervertices and superedges are active in different operational scenarios. Example 10 (Shift-Based Communication in a Global Corporation as a Soft Directed 3-SuperHyperGraph). Consider a multinational company whose employees form teams, departments, and divisions. We model information flow through three hierarchical levels of “supervertices” and allow soft selection by work shift. Base set of employees: S = {Keisuke, Bob, Sakura}. Level 1 supervertices (teams): T1 = {Keisuke,Bob}, T2 = {Bob,Sakura}, so POWS1(S) ⊇ V1 = {T1, T2}. Level 2 supervertices (departments): D1 = {T1}, D2 = {T2}, so POWS2(S) ⊇ V2 = {D1, D2}. Level 3 supervertices (divisions): U1 = {D1}, U2 = {D2}, U3 = {D1, D2}, so POWS3(S) ⊇ V3 = {U1, U2, U3}. Directed 3-superedges (communication channels): E3 = { e1 = ({U1}, {U3}), e2 = ({U2}, {U3}) } . Here e1 represents Division 1 reporting to the Corporate Division U3, and e2 likewise for Division 2. Soft selection by work shift: Let the parameter set be C = {Morning, Evening}. Define A : C −→ POWS(V3), B : C −→ POWS(E3), by A(Morning) = {U1, U2, U3}, B(Morning) = {e1, e2}, A(Evening) = {U2, U3}, B(Evening) = {e2}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 13 of 27 • Morning shift: All divisions (U1, U2) and corporate (U3) are active; both channels e1, e2 operate. • Evening shift: Only Division 2 (U2) reports to corporate (U3), so only e2 is active. Hence ( DSuHG(3), C, A, B ) is a soft directed 3-SuperHyperGraph capturing how shift schedules select which hierar- chical communication channels are used in different operational contexts. Example 11 (Logistics Network as a Soft Directed 4-SuperHyperGraph). We model a four-level logistics hierarchy—from individual vehicles up to global command—under dif- ferent operational modes. Level 0 (vehicles): S0 = {TruckA, TruckB, TruckC}. Level 1 (regional hubs): H1 = {TruckA,TruckB}, H2 = {TruckB,TruckC}, so POWS1(S0) ⊇ V1 = {H1, H2}. Level 2 (national centers): N1 = {H1}, N2 = {H2}, so POWS2(S0) ⊇ V2 = {N1, N2}. Level 3 (continental divisions): C1 = {N1}, C2 = {N2}, C3 = {N1, N2}, so POWS3(S0) ⊇ V3 = {C1, C2, C3}. Level 4 (global command): G1 = {C1, C2}, G2 = {C3}, so POWS4(S0) ⊇ V4 = {G1, G2}. Directed 4-superedges (reporting channels): E4 = { e1 = ( {G1}, {G2} ), e2 = ( {G2}, {G1} ) } . Here e1 models reports from the union of divisions to the consolidated command G2, and e2 allows feedback. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 14 of 27 Operational modes (parameters): C = {Standard, NightShift, Weekend}. Define soft selections A : C −→ POWS(V4), B : C −→ POWS(E4), by A(Standard) = {G1, G2}, B(Standard) = {e1, e2}, A(NightShift) = {G1}, B(NightShift) = {e1}, A(Weekend) = {G2}, B(Weekend) = {e2}. • Standard: Both global nodes active; both reporting and feedback channels open. • NightShift: Only the primary command node G1 operates, sending data upward via e1. • Weekend: Only the consolidated command node G2 remains active, issuing feedback via e2. Therefore, the quadruple ( DSuHG(4), C, A, B ) is a soft directed 4-SuperHyperGraph that captures how different operational modes select which global reporting channels are in use. Theorem 1 (Generalization of Directed and Soft n-SuperHyperGraphs). The notion of soft directed n-SuperHyperGraph simultaneously extends: (a) Every directed n-SuperHyperGraph DSuHG(n) = (V,E) by taking a singleton param- eter set C = {∗} with A(∗) = V and B(∗) = E. (b) Every soft n-SuperHyperGraph ( V,E, C, A, B ) by viewing each undirected superedge e ∈ E as a degenerate directed edge (e, e) ∈ E′, and setting E′ = {(e, e) : e ∈ E}. Proof. (a) Recovering directed n-SuHG. Given DSuHG(n) = (V,E), set C = {∗}, A(∗) = V , and B(∗) = E. Then by definition each soft directed sub-n-superhypergraph at ∗ is (V,E), so we exactly recover the original directed structure. (b) Recovering soft n-SuHG. Let ( V,E,C,A,B ) be a soft n-SuperHyperGraph. Define a directed n-SuperHyperGraph DSuHG′(n) with vertex set V and directed edge set E′ = { (e, e) : e ∈ E}, i.e. each undirected superedge is treated as a looped directed edge. Retain the same parameter set C and mappings A,B. Then for each c ∈ C, B(c) ⊆ { e ∈ E : e ⊆ A(c)} ⇐⇒ { (e, e) : e ∈ B(c)} ⊆ { (e, e) ∈ E′ : e ⊆ A(c)}, T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 15 of 27 so ( A(c), B(c) ) coincides with the soft directed substructure at c. Hence the soft directed n-SuperHyperGraph reduces to the original soft n-SuperHyperGraph when all edges are looped. Theorem 2. Let ( ∆∗, C, R, S ) be a soft directed hypergraph on a simple directed hypergraph ∆∗ = (Γ,Ξ), with R : C → POWS(Γ), S : C → POWS(Ξ). Define n = 1 and form the directed 1-SuperHyperGraph DSuHG(1) = (V,E) by V = { {v} : v ∈ Γ } ⊆ POWS1(Γ), E = { ( {T (e)}, {H(e)} ) : e = (T (e), H(e)) ∈ Ξ } , where each original hyperarc e becomes a directed 1-superedge between the singleton sets of its tail and head. Then the maps A : C −→ POWS(V ), A(c) = {{v} : v ∈ R(c)}, B : C −→ POWS(E), B(c) = { ({T (e)}, {H(e)}) : e ∈ S(c) } turn ( DSuHG(1), C, A, B ) into a soft directed 1-SuperHyperGraph that is isomorphic to the original soft directed hypergraph ( ∆∗, C,R, S ) . Proof. We verify each requirement in turn. 1. Construction of DSuHG(1). Since POWS1(Γ) = POWS(Γ), the set of all singleton subsets {{v} : v ∈ Γ} is a valid choice for V . Each hyperarc e = (T (e), H(e)) ∈ Ξ becomes a directed 1-superedge ({T (e)}, {H(e)}) in E ⊆ POWS1(Γ)×POWS1(Γ). By simplicity of ∆∗, no two hyperarcs share both the same tail and head, so no parallel superedges occur. 2. Definition of soft selections. For each parameter c ∈ C, set A(c) = {{v} : v ∈ R(c)} ⊆ V, B(c) = {({T (e)}, {H(e)}) : e ∈ S(c)} ⊆ E. Because S(c) ⊆ Ξ and R(c) ⊆ Γ, these indeed select subsets of V and E. 3. Validity of the induced substructure. By the soft directed hypergraph definition, every hyperarc e ∈ S(c) has T (e) ⊆ R(c) and H(e) ⊆ R(c). Therefore {T (e)} ⊆ {{v} : v ∈ R(c)} = A(c), {H(e)} ⊆ A(c), so each directed 1-superedge ({T (e)}, {H(e)}) lies in { (τ, h) ∈ E : τ, h ⊆ A(c)}. Thus (A(c), B(c)) is a valid soft directed 1-SuperHyperGraph substructure. 4. Isomorphism with the original. Define ΦV : Γ −→ V, ΦV (v) = {v}, T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 16 of 27 ΦE : Ξ −→ E, ΦE ( (T,H) ) = ({T}, {H}). Together, (ΦV ,ΦE) is a bijection of directed hypergraphs that carries the soft selection (R(c), S(c)) precisely to (A(c), B(c)). In particular ΦV ( R(c) ) = A(c), ΦE ( S(c) ) = B(c), for all c ∈ C. Hence the two soft structures are isomorphic, completing the proof. Theorem 3 (Isomorphism Invariance). Let DSuHG(n) = (V,E) and DSuHG′(n) = (V ′, E′) be two directed n-SuperHyperGraphs. Suppose φ : V → V ′ is a bijection such that for every directed superedge e = (T,H) ∈ E, (φ(T ), φ(H)) ∈ E′, where φ(T ) = {φ(v) : v ∈ T}. Given a soft structure (C,A,B) on the first graph, define A′(c) = φ ( A(c) ) , B′(c) = { (φ(T ), φ(H)) : (T,H) ∈ B(c)}. Then ( DSuHG(n), C,A,B ) ∼= ( DSuHG′(n), C,A′, B′) as soft directed n-SuperHyperGraphs. Proof. We break the proof into three parts: 1. Extend φ to supervertices. Since V ⊆ POWSn(S), define φn : POWSn(S) −→ POWSn(S′), φn(X) = {φ(v) : v ∈ X}. By bijectivity of φ, φn is also a bijection on n-supervertices. 2. Preserve directed edges. Take any directed superedge e = (T,H) ∈ E. By hypothesis, (φ(T ), φ(H)) ∈ E′. Thus φn carries the entire relation E onto E′. 3. Map soft selections. For each parameter c ∈ C: A′(c) = φn ( A(c) ) and B′(c) = {(φn(T ), φn(H)) : (T,H) ∈ B(c)}. Because φn is bijective and respects inclusion, T ⊆ A(c) ⇐⇒ φn(T ) ⊆ φn ( A(c) ) = A′(c), and similarly for heads. Hence (A(c), B(c)) is carried exactly to (A′(c), B′(c)), verifying the isomorphism of soft structures. Theorem 4 (Recovery of Directed n-SuperHyperGraphs). If C = {∗} and we define A(∗) = V, B(∗) = E, then the soft directed n-SuperHyperGraph ( DSuHG(n), C,A,B ) reduces to the underlying directed n-SuperHyperGraph DSuHG(n). T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 17 of 27 Proof. With exactly one parameter, the only soft sub-n-superhypergraph is (A(∗), B(∗)) = (V,E). There is no restriction or filtering: all supervertices and superedges are included. Therefore the soft structure coincides with the original directed graph. Theorem 5 (Recovery of Soft n-SuperHyperGraphs). Let ( V,E,C,A,B ) be a soft (undi- rected) n-SuperHyperGraph. Construct a directed looped version DSuHG′(n) = (V,E′) by E′ = {(e, e) : e ∈ E}. Then ( DSuHG′(n), C,A,B ) as a soft directed n-SuperHyperGraph recovers ( V,E,C,A,B ) . Proof. Each undirected superedge e ∈ E becomes a directed hyperedge (e, e) that loops from e back to e. For each c ∈ C: B(c) ⊆ { e ∈ E : e ⊆ A(c)} is equivalent to B(c) ⊆ { (e, e) ∈ E′ : Tail(e, e) = e ⊆ A(c)}, since Tail(e, e) = Head(e, e) = e. Hence the soft substructure (A(c), B(c)) is unchanged, demonstrating exact recovery of the original soft n-SuperHyperGraph. Theorem 6 (Monotonicity under Parameter Inclusion). If c1, c2 ∈ C satisfy A(c1) ⊆ A(c2) and B(c1) ⊆ B(c2), then for every directed superedge (T,H) ∈ B(c1), we have (T,H) ∈ B(c2), and every supervertex in A(c1) belongs to A(c2). In other words, the soft sub-n-superhypergraph at c1 is contained in that at c2. Proof. By hypothesis: 1. Every v ∈ A(c1) also lies in A(c2). 2. Every (T,H) ∈ B(c1) satisfies T,H ⊆ A(c1) ⊆ A(c2), so (T,H) is a valid element of B(c2). Therefore the inclusion of soft selections is preserved, and the theorem follows imme- diately. Theorem 7 (Intersection Closure). Let( DSuHG(n), C, A, B ) be a soft directed n-SuperHyperGraph. For any two parameters c1, c2 ∈ C, define A∧ = A(c1) ∩ A(c2), B∧ = B(c1) ∩ B(c2). Then the pair (A∧, B∧) satisfies B∧ ⊆ { e ∈ E : Tail(e) ⊆ A∧, Head(e) ⊆ A∧ } , so ( A∧, B∧ ) is itself a valid soft induced sub–n-superhypergraph. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 18 of 27 Proof. By definition of a soft directed n-SuperHyperGraph, for each i = 1, 2, B(ci) ⊆ { e ∈ E : Tail(e) ⊆ A(ci), Head(e) ⊆ A(ci)}. Hence any e ∈ B∧ = B(c1) ∩B(c2) satisfies Tail(e) ⊆ A(c1) and Tail(e) ⊆ A(c2), so Tail(e) ⊆ A(c1) ∩ A(c2) = A∧. An identical argument applies to Head(e). Therefore e lies in { e ∈ E : Tail(e) ⊆ A∧,Head(e) ⊆ A∧}, establishing the claimed inclusion. Theorem 8 (Union Closure). With the same notation, define A∨ = A(c1) ∪ A(c2), B∨ = B(c1) ∪ B(c2). Then every e ∈ B∨ satisfies Tail(e) ⊆ A∨, Head(e) ⊆ A∨, so ( A∨, B∨ ) also defines a soft induced sub–n-superhypergraph. Proof. Take any hyperedge e ∈ B∨. By definition, e belongs either to B(c1) or to B(c2). - If e ∈ B(c1), then Tail(e) ⊆ A(c1) ⊆ A(c1)∪A(c2) = A∨, and similarly Head(e) ⊆ A∨. - If e ∈ B(c2), the same argument shows Tail(e) ⊆ A∨ and Head(e) ⊆ A∨. In either case, e lies in the set of directed n-superedges whose tail and head both lie inside A∨. Consequently, B∨ ⊆ { e ∈ E : Tail(e) ⊆ A∨, Head(e) ⊆ A∨}, as required. Theorem 9 (Lattice Structure of Soft Selections). The collection of pairs {(A(c), B(c)) : c ∈ C}, ordered by (A1, B1) ⪯ (A2, B2) ⇐⇒ A1 ⊆ A2 and B1 ⊆ B2, forms a bounded lattice under the meet (∩,∩) and join (∪,∪) operations defined above. Proof. We must show that any two elements (A(c1), B(c1)) and (A(c2), B(c2)) have a greatest lower bound (meet) and a least upper bound (join) with respect to ⪯. Meet. By Theorem 7, ( A(c1) ∩A(c2), B(c1) ∩B(c2) ) is a valid soft substructure, and clearly it is the largest pair below both (A(c1), B(c1)) and (A(c2), B(c2)). T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 19 of 27 Join. By Theorem 8, ( A(c1) ∪A(c2), B(c1) ∪B(c2) ) is valid, and it is the smallest pair above both initial elements. Bounds. The minimum element is (∅,∅), since no edges or vertices lie outside the empty set. The maximum element is (V,E), the full directed n-superhypergraph. Hence the family forms a bounded lattice. Definition 11 (Acyclic Directed n-SuperHyperGraph). A directed n-SuperHyperGraph DSuHG(n) = (V,E) is acyclic if there exists no finite sequence of distinct hyperedges e1 = (T1, H1), e2 = (T2, H2), . . . , ek = (Tk, Hk) ∈ E and vertices v0, v1, . . . , vk = v0 ∈ V such that for each i = 1, . . . , k: vi−1 ∈ Ti and vi ∈ Hi. Theorem 10 (Acyclicity Preservation). Let ( DSuHG(n), C,A,B ) be a soft directed n- SuperHyperGraph whose underlying DSuHG(n) is acyclic. Then for any parameter c ∈ C, the induced soft sub-n-superhypergraph ( A(c), B(c) ) is also acyclic. Proof. Assume, for contradiction, that some soft sub-n-superhypergraph (A(c), B(c)) contains a directed cycle. Then there exist distinct edges e1, . . . , ek ∈ B(c) and vertices v0, v1, . . . , vk = v0 ∈ A(c) with vi−1 ∈ Tail(ei), vi ∈ Head(ei), i = 1, . . . , k. Since B(c) ⊆ E and A(c) ⊆ V , the same sequence of edges and vertices witnesses a cycle in the full graph DSuHG(n). This contradicts its acyclicity. Hence no such cycle can occur in any soft substructure. Definition 12 (Reachability). In a directed n-SuperHyperGraph (V,E), a vertex u ∈ V is reachable from v ∈ V if there exists a finite sequence of hyperedges e1 = (T1, H1), e2 = (T2, H2), . . . , em = (Tm, Hm) and vertices v = v0, v1, . . . , vm = u such that for each i: vi−1 ∈ Ti, vi ∈ Hi. Theorem 11 (Reachability Restriction). Under the same hypothesis, if u is reachable from v in the soft sub-n-superhypergraph ( A(c), B(c) ) , then u is also reachable from v in the original directed n-SuperHyperGraph. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 20 of 27 Proof. Suppose in (A(c), B(c)) there is a path v = v0 e1−→ v1 e2−→ · · · em−−→ vm = u, with each ei ∈ B(c) and vi−1 ∈ Tail(ei), vi ∈ Head(ei). Since B(c) ⊆ E, every ei also belongs to the full edge set E. Moreover, all vi ∈ A(c) ⊆ V . Thus the same sequence of edges and vertices forms a valid reachability path in the underlying DSuHG(n). Therefore reachability in any soft induced subgraph implies reachability in the original graph. 4. Discussion In this section, we examine Soft Directed n-SuperHyperGraphs in relation to estab- lished models. Table 1 compares Soft Directed Graphs, Soft Directed HyperGraphs, and Soft Directed n-SuperHyperGraphs, while Table 2 contrasts Soft n-SuperHyperGraphs with Soft Directed n-SuperHyperGraphs. Concept Base Structure Selection Maps Soft Directed Graph Directed graph ( V, E ) where E ⊆ V × V Each parameter chooses a subset of vertices in V and a subset of directed edges in E. Soft Directed Hy- perGraph Directed hypergraph ( Γ, Ξ ) where Ξ ⊆ P(Γ) × P(Γ) Each parameter chooses a subset of nodes in Γ and a subset of hyperarcs in Ξ. Soft Directed n-SuperHyperGraph Directed n-SuperHyperGraph( V, E ) with V ⊆ Pn(S), E ⊆ Pn(S) × Pn(S) Each parameter chooses a subset of n-supervertices in V and a subset of directed n-superedges in E. Table 1: Overview of soft-parameterized directed network models Concept Base Structure Selection Maps Soft n-SuperHyperGraph n-SuperHyperGraph ( V, E ) with V ⊆ Pn(S), E ⊆ Pn(S) Each parameter chooses a subset of n-supervertices in V and a subset of n-superedges in E. Soft Directed n-SuperHyperGraph Directed n-SuperHyperGraph( V, E ) with V ⊆ Pn(S), E ⊆ Pn(S) × Pn(S) Each parameter chooses a subset of directed n-supervertices in V and a subset of directed n-superedges in E. Table 2: Overview of Soft n-SuperHyperGraphs versus Soft Directed n-SuperHyperGraphs From these comparisons, it is clear that Soft Directed n-SuperHyperGraphs excel at modeling highly hierarchical, complex, and inherently directional phenomena. Future work involving computational experiments and algorithm development for Soft Directed n-SuperHyperGraphs will yield both qualitative insights and detailed quantitative evalu- ations, paving the way for practical applications. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 21 of 27 5. Conclusion and Future Tasks In this paper, we explored the mathematical properties and real-world applications of Soft Directed n-SuperHyperGraphs. These structures provide a flexible and expressive framework for modeling hierarchical, directional, and parameterized relationships under uncertainty. As directions for future work, we aim to investigate further generalizations and exten- sions by incorporating advanced uncertainty-based structures such as HyperSoft Sets[31], Soft Expert Sets [69], Hypersoft Expert Set[70–72], SuperHyperSoft Sets[73, 74], and Soft Rough Sets [75]. These developments have the potential to enhance modeling capabilities in complex and imprecise domains. Furthermore, several more refined notions of directed graphs have been proposed in the literature, such as bidirected graphs [76, 77], bunch graphs [78], pangene graphs[79], and multidirected graphs [80]. As part of future work, we intend to explore potential extensions of our framework using these advanced graph structures. Furthermore, we also hope that future research will explore applications of Soft Directed n-SuperHyperGraphs in areas such as decision-making, network theory, and graph neural networks. Acknowledgments The authors extend their sincere thanks to all who offered valuable insights, feedback, and encouragement. We appreciate our readers’ interest and the foundational contribu- tions of the scholars whose work underpins this study. We also acknowledge the institutions and individuals who provided the necessary infrastructure and support. Finally, we are grateful to everyone who helped make this research possible. Data Availability This work is purely theoretical and does not involve original datasets. We invite future empirical research to validate and expand upon the ideas presented here. Ethical Approval No ethical review was required, as this study did not involve human participants or animal subjects. Conflicts of Interest The authors declare that they have no competing interests related to this publication. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 22 of 27 Funding This research received no financial support from any external organization or funding body. Research Integrity We confirm that this manuscript reports original work by the authors, has not been published elsewhere, and is not under consideration by another journal. Disclaimer on Computational Tools All proofs and analyses were performed manually. We did not employ any computer algebra systems, automated theorem provers, or proof assistants (e.g., Mathematica, Sage- Math, Coq). Code Availability No software or source code was produced for this theoretical investigation. Clinical Trial This study did not include any clinical trials. Consent to Participate Not applicable. Disclaimer on Scope and Accuracy The concepts and models introduced here have yet to undergo empirical testing. Future work may explore their practical applications and rigorously assess their validity. While we have strived for accuracy and proper citation, inadvertent errors may remain; readers are encouraged to consult original sources and contact us with any questions. The conclusions are drawn within the specific theoretical framework and assumptions described, and may not generalize beyond those boundaries. All viewpoints expressed are those of the authors alone. Consent to Publish All authors have given their consent for submission of this manuscript to the journal. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6643 23 of 27 References [1] Reinhard Diestel. Graph theory. Springer (print edition); Reinhard Diestel (eBooks), 2024. [2] Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984. [3] Florentin Smarandache. 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